Reciprocal relationships, reverse causality, and temporal ordering

Testing theories with cross-lagged panel models

Charles C. Lanfear

University of Cambridge

Background

What is today about?

Reviewing common problems in applied work

Aligning theory, models, and estimation

Assume understand basics of conditional ignorability

Reciprocality

  • Theories sometimes imply reciprocality
  • Also sometimes an empirical problem
  • But pretty much nothing is actually reciprocal; the world is recursive

Mechanisms and time

  • Time enforces recursiveness
  • Things that appear non-recursive are usually omitting mediators or time

Show 5 time DAG, 3 time SEM

Do another with a mechanism

Time DAG

Figure 1: ?(caption)

Time Estimator

Theory

  • Make your theory recursive; use a DAG
  • Estimation may requite non-recursive models

Three common problems

Temporal Order

  • Illustrated by Vaisey & Miles (2019)

  • If true model is \(y = \beta x_t + \alpha_i + e_{it}\) and you fit \(y = \beta^* x_{t-t} + \alpha_i + e_{it}\), \(E(\beta^*) = -0.5\beta\)

Example paper

Solutions

  • Hard: Use strong theory to get timing right
  • Use robust estimators, e.g., Allison#s approach
  • If ambiguous contemporaneous path matters, you’ll need strong instrument(s)

Unobserved Heterogeneity

  • Most common concern is stable traits
  • Can’t just toss in fixed effects due to Nickell bias

Example paper

Solutions

  • Different approaches
  • Psychology approaches: RI-CLPM
  • Econometric approaches: Arellano-Bond, ML-SEM

Low Temporal Variation

  • \(Var(Y_2|Y_1) \rightarrow 0\) as \(\rho(Y_1,Y_2) \rightarrow 1\)
  • Error becomes proportionally larger component
  • Common with narrow waves, stable constructs

Example paper

That neighbs one? Hipp?

Solutions

  • Error part can be dealt with using measurement models
    • Should do this anyway as outcomes are regressors, so measurement error attenuates estimates
  • Address during data collection
    • Oversample for change
    • Look for shocks
  • Consider different time lags or units of analysis
  • Might be nothing you can do

Advice

  • Theory comes first
  • Default to robust methods

Feedback and Questions

Contact:

Charles C. Lanfear
Institute of Criminology
University of Cambridge
cl948@cam.ac.uk